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On uniform relationships between combinatorial problems
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Dorias, FG, Dzhafarov, DD, Hirst, JL et al. (2 more authors) (2016) On uniform
relationships between combinatorial problems. Transactions of the American Mathematical
Society, 368 (2). pp. 1321-1359. ISSN 0002-9947
https://doi.org/10.1090/tran/6465
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PROBLEMS
FRANC¸ OIS G. DORAIS, DAMIR D. DZHAFAROV, JEFFRY L. HIRST, JOSEPH R. MILETI, AND PAUL SHAFER
Abstract. The enterprise of comparing mathematical theorems according to their logical strength is an active area in mathematical logic, with one of the most common frameworks for doing so being reverse mathematics. In this set-ting, one investigates which theorems provably imply which others in a weak formal theory roughly corresponding to computable mathematics. Since the proofs of such implications take place in classical logic, they may in principle involve appeals to multiple applications of a particular theorem, or to non-uniform decisions about how to proceed in a given construction. In practice, however, if a theorem Qimplies a theorem P, it is usually because there is a direct uniform translation of the problems represented byPinto the prob-lems represented byQ, in a precise sense formalized by Weihrauch reducibility. We study this notion of uniform reducibility in the context of several natural combinatorial problems, and compare and contrast it with the traditional no-tion of implicano-tion in reverse mathematics. We show, for instance, that for all n, j, k≥1, ifj < kthen Ramsey’s theorem forn-tuples and kmany colors is not uniformly, or Weihrauch, reducible to Ramsey’s theorem forn-tuples andjmany colors. The two theorems are classically equivalent, so our anal-ysis gives a genuinely finer metric by which to gauge the relative strength of mathematical propositions. We also study Weak K¨onig’s Lemma, the Thin Set Theorem, and the Rainbow Ramsey’s Theorem, along with a number of their variants investigated in the literature. Weihrauch reducibility turns out to be connected with sequential forms of mathematical principles, where one wishes to solve infinitely many instances of a particular problem simultane-ously. We exploit this connection to uncover new points of difference between combinatorial problems previously thought to be more closely related.
1. Introduction
The idea of reducing, or translating, one mathematical problem to another, with the aim of using solutions to the latter to obtain solutions to the former, is a basic and natural one in all areas of mathematics. For instance, the convolution of two functions can be reduced to a pointwise product via the Fourier transform;
Dzhafarov was partially supported by an NSF Postdoctoral Fellowship. Hirst was partially supported by grant ID#20800 from the John Templeton Foundation. (The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation.) Shafer was supported by the Fondation Sciences Math´ematiques de Paris and is also an FWO Pegasus Long Postdoctoral researcher. We are grateful to D. Hirschfeldt and C. Jockusch for numerous helpful comments and discussions during the preparation of this article, and for a remark that helped strengthen Proposition4.7. We thank J. Miller for pointing out Kummer’s theorem, Theorem5.16, to us. We also thank V. Brattka, A. Montalb´an, A. Marcone, C. Mummert, and the anonymous referee for bringing to our attention the connections of our work with Weihrauch reducibility, of which we were initially unaware. We additionally thank the referee for a number of other useful comments.
the study of a linear operator over a complex vector space can be reduced to the study of a matrix in Jordan normal form, via a change of basis; etc. In general, the precise forms of such reductions vary greatly with the particular problems, but they tend to be most useful when they are constructive or uniform in some appropriate sense. Typically, such reductions preserve various fundamental properties and yield more information, and they are usually easier to implement. These ideas have materialized in many areas such as category theory, complexity theory, proof theory, and set theory (see [3]). In this article, we investigate similar uniform reductions between various combinatorial problems in the setting of computability theory, reverse mathematics and computable analysis.
The program of reverse mathematics provides a unified and elegant way to com-pare the strengths of many mathematical theorems. Its setting is second-order arithmetic, which is a system strong enough to encompass most of classical math-ematics. The formalism permits talking about natural numbers and about sets of natural numbers, and hence readily accommodates countable analogues of math-ematical propositions. The fundamental idea is to calibrate the proof-theoretical strength of such propositions by classifying which set-existence axioms are needed to establish the structures needed in their proofs. In practice, we work with frag-ments, or subsystems, of second-order arithmetic, first finding the weakest one that suffices to prove a given theorem, and then obtaining sharpness by showing that the theorem is in fact equivalent to it. Each of the subsystems corresponds to a natural closure point under logical, and more specifically, computability-theoretic, opera-tions. Thus, the base system, Recursive Comprehension Axiom (RCA0), roughly corresponds to computable or constructive mathematics; the system Weak K¨onig’s Lemma (WKL0) corresponds to closure under taking infinite paths through infinite binary trees; and the Arithmetical Comprehension Axiom (ACA0) corresponds to closure under arithmetical definability, or equivalently, under applications of the Turing jump. Other common subsystems, ATR0 and Π1
1-CA0, which we shall not
consider in this article, admit similar characterizations. The point is that there is a rich interaction between proof systems on the one hand, and computability on the other.
We refer the reader to Simpson [29] for background on reverse mathematics, to Soare [30] for background on computability theory, and to Weihrauch [35] for background in computable analysis. For background on algorithmic randomness, to which some of our results in Sections4 and6 will pertain, we refer to Downey and Hirschfeldt [13].
In the context of reverse mathematics, we can say that a theorem P“reduces” to a theoremQif there is a proof of PassumingQoverRCA0. Since these proofs are carried out in a formal system, such a proof of P from Q may use Q several times to obtainP, or may involve non-uniform decisions about which sets to use in a construction. However, in many natural cases, a proof of P from Q uses direct, computable, and uniform translations between problems represented byP
into problems represented byQ.
To describe these types of arguments more precisely, we restrict our focus to Π1 2
statements in the language of second-order arithmetic, i.e., statements of the form
whereϕis arithmetical. Each such principle has associated to it a natural class of instances, and for each instance, a natural class of solutions to that instance. The following are a few important examples.
Statement 1.1 (WKL). Every infinite subtree of 2<ω has an infinite path.
Statement 1.2 (WWKL). Every subtree T of 2<ω such that
|{σ∈2n:σ∈T}|
2n
is uniformly bounded away from zero for allnhas an infinite path.
Statement 1.3 (Ramsey’s Theorem). Fixn, k≥1. RTnk is the statement that for
every f: [ω]n →k, there exists an infinite setH (calledhomogeneous forf) such
thatf is constant on [H]n.
Statement 1.4 (COH). For every sequence of sets hRi : i ∈ ωi, there exists an infinite setC such that for alli, eitherC∩Ri is finite orC∩Ri is finite.
The idea of uniform direct translations alluded to above was made precise by Weihrauch [33,34] in the realm of computable analysis and has been widely studied ever since (see [5, Section 1] for a partial bibliography). In this context, a Π1 2
statement (∀X)(∃Y)ϕ(X, Y) as above is viewed as a function specification; a partial realizer of such a specification is a function F such that ϕ(X, F(X)) holds for all X in the domain of F. For this reason, it is traditional in this context to understand the relationϕ(X, Y) as a partial multi-valued function whereϕ(X, Y) holds exactly whenY is one of the possible values of the function atX. Weihrauch then introduced a notion of computable reducibility between partial multi-valued functions whereby there are computable processes that serve to uniformly translate realizers of one partial valued function into realizers of another partial multi-valued function. We shall use here the following equivalent definition, which may appear more familiar from perspectives outside of computable analysis, particularly reverse mathematics. However, with a view towards encouraging more collaboration between these two similary-motivated but thus far largely separate approaches, we include an equivalence of the definitions in AppendixA.
Definition 1.5. Let P and Q be Π1
2 statements of second-order arithmetic. We
say that
(1) P is Weihrauch reducible to Q, and write P ≤W Q, if there exist Turing
reductions Φ and Ψ such that wheneverAis an instance ofPthenB = Φ(A) is an instance ofQ, and wheneverT is a solution toB thenS= Ψ(A⊕T) is a solution toA.
(2) Pis strongly Weihrauch reducible to Q, and writeP≤sW Q, if there exist
Turing reductions Φ and Ψ such that wheneverAis an instance ofPthen
B = Φ(A) is an instance of Q, and whenever T is a solution to B then
S= Ψ(T) is a solution toA.
In other words, Weihrauch reducibility differs from strong Weihrauch reducibility only in that the “backwards” reduction Ψ takes as oracle not only the solutionT
Weihrauch and related reducibilities in the context of computable combinatorics. We note that the notation in these sources differs from ours, with≤uand≤subeing
used in place of≤Wand≤sW, respectively.) For most of our results below (with the
notable exception of Theorem 3.1) it will not matter which of the two reducibility notions we are working with, so to present the strongest possible results, we shall prove reductions for≤sW, and non-reductions forW.
It is straightforward to see each of these reducibilities is reflexive and transitive and thus defines a degree structure on Π1
2 statements.
One simple example of a strong Weihrauch reduction is thatRTnj ≤sWRTnk
when-everj ≤k because given f: [ω]n →j, we may viewf as a function g: [ω]n → k
(by ignoring the additional colors) and then every set homogeneous for g is ho-mogeneous for f. (Thus, here Φ and Ψ can both be taken to be the identity reduction.) A slightly more interesting example is that RTmk ≤sW RTnk
when-ever m ≤ n. To see this, given f: [ω]m → k, define g: [ω]n → k by letting g(x1, . . . , xm, . . . , xn) =f(x1, . . . , xm) and notice thatgis uniformly obtained from
f via a Turing functional, and that every set homogenous forgis homogeneous for
f.
There are also many examples of such reductions using more complicated Turing functionals. Friedman, Simpson, and Smith [16] showed that if Pis the statement that every commutative ring with identity has a prime ideal, thenRCA0⊢WKL→
P. Adapting the proof of this result, one can show that it is possible to uniformly computably convert a commutative ring Rinto an infinite tree T such that every path ofT is a prime ideal ofR, and hence thatP≤sWWKL. For another
exam-ple, Cholak, Jockusch, and Slaman [9, Theorem 12.5] exhibit a strong Weihrauch reduction ofCOH toRT22via a non-trivial Φ.1
Despite the fact that many natural implications in reverse mathematics corre-spond to Weihrauch reductions (even strong Weihrauch reductions), there are cer-tainly examples where an implication holds in reverse mathematics but no Weihrauch reduction exists. For example, building on work of Jockusch in [21], it is known that RCA0 ⊢ RTnk ↔ ACA whenever n ≥ 3 and k ≥ 2, and in particular that RCA0 ⊢ RT32 →RT42. However, RT42 W RT32 because every computable instance
of RT32 has a ∅′′′-computable solution, but there is a computable instance of RT42
with no∅′′′-computable solution (see [21, Theorems 5.1 and 5.6]). The underlying
reason why this implication holds in reverse mathematics is that ∅′ can be coded
into a computable instance of RT32, and by relativizing and iterating this result (i.e., by using multiple nested applications of RT32), one can obtain the several jumps necessary to compute solutions to instances ofRT42.
1The same is not true ifRT2
2 is replaced by the closely related principleD 2
2, introduced in [9, Statement 7.8]. This asserts that iff : [ω]2
→2 isstable, i.e., if for eachxthe limit off(x, y) as y tends to infinity exists, then there is an infinite set consisting either entirely of numbers for which this limit is 0, or entirely of numbers for which this limit is 1. A recent result by Chong, Slaman, and Yang [10, Theorem 2.7] resolves a longstanding open problem by showing that RCA0 0 D22 → COH. However, the model they construct to witness the separation is a non-standard one, and so leaves open the question of whether every ω-model of RCA0+D22 is also a model ofCOH. A typical reason for this being the case would be ifCOHwere uniformly reducible toD2
2. However, by results of Dzhafarov [14, Theorem 1.5 and Corollary 1.10] it follows thatCOHsWD22, and more recently, Lerman, Solomon, and Towsner (unpublished) have shown even thatCOHWD
There are also more subtle instances where no Weihrauch reduction exists de-spite the fact that degrees of solutions to the problems correspond. For example, Jockusch [22, Theorem 6] showed that for anyk≥2, the degrees of DNRk functions
(i.e., functionsf:ω →ksuch that f(e)6= Φe(e) for alle∈ω) are the same as the
degrees of DNR2 functions, but there is no Weihrauch reduction witnessing this.
More precisely, he showed that givenk ≥2, there is no Turing functional Φ such that Φ(g)∈DNRk for allg ∈DNRk+1. If we let DNRk be the Π12 statement “for
every X, there exists a DNRk function relative to X”, then Jockusch’s theorem
shows thatDNRk W DNRk+1.
A motivating question for this article is what happens when one varies the num-ber of colors in Ramsey’s Theorem. It is well known that if n≥ 1 and j, k ≥2, thenRCA0⊢RTnj ↔RTnk. For example, to see thatRCA0 ⊢RTn2 →RTn3, we can
argue as follows. Suppose thatf: [ω]n→3. Defineg: [ω]n →2 by letting
g(x) =
(
0 iff(x)∈ {0,1},
1 iff(x) = 2.
ByRTn2, we may fix a setH such thatHis homogenous forg. Now ifg([H]2) ={1},
then H is homogeneous for f. Otherwise, the function f ↾[H]2 is a 2-coloring of
[H]2, so we may apply RTn
2 a second time to conclude that there is an infinite I⊆H such thatIis homogeneous forf. Notice that this proof requires two nested applications of RTn2 to obtain a solution to RTn3. However, there are no known degree-theoretic differences between homogeneous sets of computable instances of
RTn2 and homogeneous sets of computable instances ofRTn3, so it is unclear whether
there is a proof of RTn3 using one uniform application of RTn2. We prove below in
Theorem3.1that RTnk sWRTnj whenj < k.
Although the same basic idea of (strong) Weihrauch reducibility is used in the contexts of computable combinatorics, computable analysis, and reverse mathemat-ics, there are important differences beyond terminology that the reader should keep in mind when translating back and forth.
• In [5], (strong) Weihrauch reduction is defined not only for partial multi-valued functions but also for abstract collections of partial functions on Baire space. This more general idea has no equivalent formulation in second-order arithmetic and, moreover, only definable relations make sense in the latter context. Thus, reverse mathematics has a limited view of the (strong) Weihrauch degrees considered in computable analysis. In prac-tice, this limitation only surfaces when considering the general structure of (strong) Weihrauch degrees since these degrees, when considered for their own sake, generally correspond to definable relations. In this paper, we will only consider arithmetically-definable relations, which therefore make sense in all contexts.
• Computable combinatorics and computable analysis work exclusively with the standard natural numbers whereas reverse mathematics also considers non-standard models. Since the base systemRCA0 only postulates
induc-tion for Σ0
1 formulas, issues related to induction often occur in translation
and it is not the case that every reductionP≤sWQtranslates into a proof
that RCA0 ⊢ Q → P. For example, a direct analysis of the reduction of Cholak, Jockusch, and Slaman showing that COH ≤sW RT22 alluded to
above appears to use Σ0
for the transformed coloring are indeed cohesive for the given instance, and some additional work is required to verify that the proof goes throughRCA0
(see Mileti [27, Appendix A]).
• The typical use of oracles varies in the three contexts. In computable com-binatorics, results are usually stated without any use of oracles and issues of relativization are discussed where necessary. In computable analysis, both the (strong) Weihrauch reduction above and its continuous analogue, which permits the use of any oracle, are considered. In reverse mathe-matics, it is customary to allow any oracle that exists in the model under consideration. So the case of (strong) Weihrauch reduction corresponds to the minimal standard model of RCA0 where the only sets are the com-putable ones, and the continuous analogue corresponds to the case of the full standard model of second-order arithmetic where all sets are present. For these reasons, most of our results will be stated in a way that includes all relevant translations, though our proofs will generally focus only on one point of view, with the others left to the reader.
We use standard notations and conventions from computability theory and re-verse mathematics. We identify subsets of ω with their characteristic functions, and we identify each n∈ω with its set of predecessors. Lower-case letters such as
i, j, k, ℓ, m, n, x, y, . . . denote elements of ω. Given a setA⊆ω, we let [A]n denote
the set of all subsets ofA of sizen. We usex,y, . . . to denote finite subsets of ω, which we identity with the corresponding tuple listing the elements in increasing order. We writex<yif maxx<miny. Given a Turing functional Φ, we assume that if Φ(A)(x)↓, then Φ(A)(y)↓for ally≤x. We say that a Turing functional Φ istotal if Φ(A) is a total function for everyA∈2ω. Given setsAandB, we write
Φ(A, B) in place of Φ(A⊕B).
2. The Squashing Theorem and sequential forms
We can naturally combine two Π1
2 principles Pand Q into one as follows. We
define the parallel product hP,Qito be the Π1
2 principle whose instances are pairs
hA, Bisuch thatAis an instance ofPandB is an instance ofQ, and the solutions to this instance are pairshS, Tisuch thatS is a solution toAandT is a solution to B. Obviously, this can be generalized to combine any number of Π1
2principles,
even an infinite number. In particular, one of our interests will be in cases whenP
andQ are the same principle. Forα∈ω∪ {ω}, we letαapplications ofP, orPα,
refer to the Π1
2principle whose instances are sequenceshAi :i < αisuch that each Ai is an instance ofP, and the solutions to this instance are sequenceshSi:i < αi
such that each Si is a solution to Ai. The infinite casePω is sometimes known as
theparallelizationofPand is also denotedPb.
Notice that we trivially have COH2 ≤sW COH because given two sequences of
sets hRi : i ∈ωi and hSi : i ∈ ωi, we can uniformly computably interleave them to form the sequencehTi :i∈ωiwhere T2i =Ri andT2i+1 =Si, so that any set cohesive for hTi : i ∈ωi is cohesive for each of hRi : i ∈ ωi and hSi : i∈ ωi. In fact, using a pairing function, it is easy to see that COHω≤sWCOH. For another
example, we have thatWKL2≤sWWKLas follows. Given two infinite treeshT0, T1i,
and that given a path B through S, the even bits form a path through T0, and the odd bits form a path through T1. Moreover, using a pairing function again, we can interleave a sequence hTi : i ∈ ωi of infinite trees together to form one infinite tree such that from any path we can uniformly computably obtain paths through each of the original trees, and hence WKLω ≤sW WKL. (This fact is also
a consequence of Theorem 8.2 of [5], which shows thatWKLis strong Weihrauch equivalent to LLPOω; see also Lemma 5 of Hirst [20] for a formalized version in reverse mathematics.)
We have the following important example using distinct principles.
Proposition 2.1. If n, j, k≥1, thenhRTnj,RTnki ≤sWRTnjk.
Proof. Given hf, gi where f: [ω]n → j and g: [ω]n → k, define h: [ω]n → jk by h(x) =hf(x), g(x)i for allx∈[ω]n. Then his uniformly computable fromhf, gi,
and any infinite homogeneous set forhis also homogeneous for bothf andg.
Given a Π1
2 principle P, if P2 ≤sW P, then it is straightforward to see (by
repeatedly applying the reduction procedures) thatPn≤sWPfor each fixedn∈ω.
For example, ifn= 4 and we are givenhA0, A1, A2, A3iwhere eachAiis an instance ofP, then
Φ(A0,Φ(A1,Φ(A2, A3)))
is an instance of Puniformly obtained from hA0, A1, A2, A3i, and from any solu-tion to this instance we can repeatedly apply Ψ to uniformly obtain a sequence
hS0, S1, S2, S3i such that each Si is a solution to Ai. (The same is true if ≤sW
is replaced by ≤W.) It is not at all clear, however, whether this process can be
continued into the infinite, i.e., does P2 ≤sW Pnecessarily imply thatPω≤sWP?
Given a sequencehAi:i∈ωiwhere eachAi is an instance ofP, the natural idea is to consider
Φ(A0,Φ(A1,Φ(A2,Φ(A3, . . .)))).
Of course, this process clearly fails to converge and so does not actually define an instance ofP. In fact, we will see later thatP2≤sWPdoes not always imply that Pω≤sWP.
However, if P2 ≤sW P and P is reasonably well-behaved, we will prove that
such a “squashing” of infinitely many applications of P into one application of
P is indeed possible. For example, consider P= RT22. The idea is to force some
convergence in the above computation by approximating the second coordinate of Φ as follows. When attempting to simulate Φ(A0,Φ(A1,Φ(A2, . . .))), we approximate the unknown result of Φ(A1,Φ(A2, . . .)) by guessing that it starts as the all zero coloring. By assuming this and hence that the second argument looks like a string of zeros, we eventually force convergence of Φ(A0,0n) on 0, at the cost of introducing
some finite initial error in the true “computation” of Φ(A1,Φ(A2, . . .)). Since removing finitely many elements from an infinite homogenous set results in an infinite homogeneous set, these finitely many errors we have introduced into the coloring will not be a problem.
More precisely, we will define a sequence hBi : i ∈ωiof instances of P(where intuitively Bi = Φ(Ai,Φ(Ai+1,· · ·)) beyond some finite error introduced to force convergence), along with a uniformly computable sequence of numbershmi:i∈ωi, such that
Now since we no longer haveBi= Φ(Ai, Bi+1) (due to the finite error), the Turing functional Ψ may not convert a solution of Bi into a pair of solutions to Ai and
Bi+1. In order to deal effectively with these finite errors, to ensure that our Bi
are actually instances of P, and to ensure that sequencehmi :i∈ωiis uniformly computable (and hence can be used as markers for cut-off points), we need to make some assumptions aboutP.
Definition 2.2. Let P be a Π1
2 principle (or, more generally, any multi-valued
function with domain 2ω).
(1) Pistotalif every element of 2ω is (or codes) an instance ofP.
(2) Phasfinite toleranceif there exists a Turing functional Θ such that when-everB1 andB2 are instances ofPwith B1(x) =B2(x) for allx≥m, and
S1 is a solution toB1, then Θ(S1, m) is a solution toB2.
Proposition 2.3. For each n, k ≥ 1, the principle RTnk is total and has finite
tolerance.
Proof. We can view every element of 2ωas a validk-coloring through simple coding.
Define Θ as follows. Given m∈ω, compute the largest elementℓ of any tuple of [ω]n coded by a natural number less thanm, and let Θ(S, m) ={a∈S :a > ℓ}.
Now ifB1 andB2 are colorings of [ω]n usingkcolors such that B1(x) =B2(x) for
allx≥m, andS1 is an infinite set homogeneous forB1, then Θ(S1, m) is also an
infinite set and it is homogeneous forB2.
Another simple example of a total principle with finite tolerance isCOH, where in fact we may take Θ(S, m) =S (because anything cohesive for a given family of sets is also cohesive for any finite modification of that family).
Although we are certainly interested in the case where P2 ≤sW P, i.e., when
hP,Pi ≤sW P, we will need a slightly more general formulation below. As above,
whenhQ,Pi ≤sWP, it is straightforward to see thathQn,Pi ≤sW Pfor each fixed n∈ω. When passing to the infinite case, however, our “squashing” never reaches the initial instance ofP, but in good cases we can conclude thatQω≤
sWP. Notice
that ifQ=P, this reduces to the case discussed above.
Remark 2.4. As a rule, all results in this section about Π1
2 principles could be
formulated more generally for any multi-valued function with domain 2ω, as in
Definition2.2. For brevity, we shall omit repeatedly stating this.
Theorem 2.5 (Squashing Theorem). Let PandQ beΠ1
2 statements, and assume
that both are total and that Phas finite tolerance. (1) If hQ,Pi ≤sWPthenQω≤sWP.
(2) If hQ,Pi ≤WPthenQω≤WP.
Proof. We prove (1), the proof of (2) being virtually the same (in fact, the argu-ment can be made somewhat simpler because the oracle has access to the original problem). Throughout, ifσ, τ ∈2<ω, we writeστ for the concatenation ofσbyτ,
andσ⌢τ for the continuation ofσbyτ, meaning
σ⌢τ(i) =
(
σ(i) ifi <|σ|, τ(i) if|σ| ≤i <|τ|,
Fix functionals Φ and Ψ witnessing the fact thathQ,Pi ≤sWP. SincePis total,
we may fix a computable instanceCofP(one could takeCto be the sequence of all 0s, but for some particular problems it might be more convenient or natural to use a differentC). Given a sequencehAi :i∈ωiof instances ofQ, we uniformly define a sequence hBi : i ∈ ωi of instances of P together with a uniformly computable sequencehmi:i∈ωiof numbers so that
Bi = (C↾mi)⌢Φ(Ai, Bi+1)
for all i. In other words, we will have Bi(x) =C(x) for all x < mi, and Bi(x) = Φ(Ai, Bi+1)(x) for all x ≥ mi. We will then use the instance B0 of P as our transformed version of hAi : i ∈ ωi and show how given a solution T0 of B0, we can uniformly transformT0into a sequencehSi:i∈ωiof solutions tohAi:i∈ωi. One subtle but very important point here is that our sequence hmi : i ∈ ωi of cut-off positions will need to be uniformly computable independent of the instances
hAi : i ∈ ωi, so that we can use them to unravel a solution T0 of B0 without knowledge of the initial instance.
Thus, our first goal is to define the uniformly computable sequencehmi:i∈ωi. We proceed in stages, initially letting m0 = 0. At stage s, we define ms+1. The goal is to choose ms+1 large enough to ensure that all potential Bi for i≤s will be defined ons. Intuitively, by placing enough ofC down in columns+ 1 (i.e., at the beginning ofBs+1), we must eventually see convergence on previous columns through the cascade effect of the nested Φ. Since we do not have access to the sequencehAi:i∈ωi, we make essential use of compactness and the fact thatQis total to handle all potential inputs at once.
To this end, assume mt has been defined for each t ≤s. First we claim there exists ann∈ω such that for allσ0, . . . , σs∈2n,
Φ(σs, C↾n)(s)↓,
Φ(σs−1,(C↾ms)⌢Φ(σs, C↾n))(s)↓,
Φ(σs−2,(C↾ms−1)⌢Φ(σs−1,(C↾ms)⌢Φ(σs, C↾n)))(s)↓,
and for generali≤s,
(1) Φ(σi,(C↾mi+1)⌢· · ·⌢Φ(σs
−1,(C↾ms)⌢Φ(σs, C↾n))· · ·)(s)↓.
Observe that the set of all such nis closed under successor. Thus, once the claim is proved, we can definems+1 to be the least suchnthat is greater thanmtfor all
t≤sand also greater thans(to ensure thatBs+1 will be defined on 0,1, . . . , s as well). This observation also implies that to prove the claim, it suffices to fixi≤s, and prove that we can effectively find annsuch that (1) holds for allσi, . . . , σs∈2n.
To this end, let T be the set of all tuples hσi, . . . , σsi of binary strings with
|σi|=· · ·=|σs|such that
Φ(σi,(C↾mi+1)⌢· · ·⌢Φ(σs
−1,(C↾ms)⌢Φ(σs, C↾|σi|)· · ·)(s)↑.
Since each of the computations here has a finite string as an oracle, T is a com-putable set. Furthermore, if hτi, . . . , τsi is an initial segment ofhσi, . . . , σsi under component-wise extension, that is ifτiσi, . . . , τsσs, thenhτi, . . . , τsibelongs to T if hσi, . . . , σsi does. Thus, T is a subtree in (2<ω)s under component-wise
Now if T is infinite, then it must have an infinite path hUi, . . . , Usi, where
Ui, . . . , Us∈2ωandhUi↾k, . . . , Us↾ki ∈T for allk. Then by definition ofT,
Φ(Ui,(C↾mi+1)⌢· · ·⌢Φ(Us−1,(C↾ms)⌢Φ(Us, C))· · ·)(s)↑.
As PandQ are both total, each ofUi, . . . , Us are instances ofQ, and each of the second components of any Φ above are instances ofP. In particular,
(C↾mi+1)⌢· · ·⌢Φ(Us
−1,(C↾ms)⌢Φ(Us, C))
is an instance V ofP, as is Φ(Ui, V). But then Φ(Ui, V)(s) cannot be undefined. We conclude that T is finite, whence its height can clearly serve as the desired n. To complete the proof, we note that an index for T as a computable tree can be found uniformly computably fromiandm0, . . . , ms, and therefore so cann.
We now define our reduction procedures witnessing that Qω ≤
sW P. LethAi : i∈ωibe an instance ofQω. From this sequence, we uniformly computably define
a sequencehBi :i∈ωiof instances of Pas follows. Again, we proceed by stages, doing nothing at stage 0. At stages+ 1, we defineBi(s) for each i≤sand define
Bs+1 on 0,1, . . . , s. Ifs < mi, we letBi(s) =C(s). Otherwise, we let
Bi(s) = Φ(Ai,(C↾mi+1)⌢· · ·⌢Φ(As−1,(C↾ms)⌢Φ(As, C↾ms+1))· · ·)(s),
the right-hand of which we know to be convergent by definition ofms+1. That is, we have defined
Bs(s) = Φ(As, C↾ms+1)(s),
Bs−1(s) = Φ(As−1,(C↾ms)⌢Φ(As, C↾ms+1))(s),
Bs−2(s) = Φ(As−2,(C↾ms−1)⌢Φ(As−1,(C↾ms)⌢Φ(As, C↾ms+1)))(s),
and so forth. (Each of the At in the computations above could also be replaced byAt↾ms+1.) We also defineBs+1(j) =C(j) for all j≤s. Since, from the next stage on,Bs+1will be defined so thatBs+1↾ms+1=C↾ms+1, it is not difficult to see that we do indeed succeed in arrangingBi= (C↾mi)⌢Φ(Ai, Bi+1), as desired. Furthermore,hBi: i∈ωi is defined uniformly computably fromhAi :i∈ωi, and eachBi is an instance ofPbecausePis total. In particular, and there is a Turing functional that producesB0from hAi:i∈ωi.
Let Θ be a Turing functional witnessing thatP has finite tolerance. We claim that from any solution to the instanceB0ofP, we can uniformly computably obtain a sequence of solutions tohAi :i∈ωi. So supposeT0 is any such solution toB0. The idea is to repeatedly apply the reduction Θ to deal with the finite errors, followed up by Ψ to convert individual solutions to pairs of solutions. Indeed, since
B0(x) = Φ(A0, B1)(x) for all x ≥ m0, we have that Θ(T0, m0) is a solution to Φ(A0, B1). Thus, Ψ(Θ(T0, m0)) =hS0, T1iis such thatS0 is a solution toA0, and
T1 is a solution toB1. The first of these,S0, can serve as the first member of our sequence of solutions. Since B1(x) = Φ(A1, B2)(x) for all x ≥m1, we have that Θ(T1, m1) is a solution to Φ(A1, B2). Thus, Ψ(Θ(T1, m1)) = hS1, T2i is such that
S1 is a solution toA1, andT2is a solution toB2. Continuing in this way, we build an entire sequencehSi:i∈ωiof solutions tohAi:i∈ωi, and sincehmi :i∈ωiis uniformly computable, we do this uniformly computably fromT0 alone. The proof
is complete.
no (strong) Weihrauch reduction of ω instances of that principle to one, and in general, showing this tends to be easier.
Corollary 2.6. Let Pbe aΠ1
2 principle that is total and has finite tolerance.
(1) If P2≤
sWP, thenPω≤sWP.
(2) If P2≤
WP, thenPω≤W P.
Proof. Apply Theorem2.5withQ=P.
Lemma 2.7. Let PandQbe Π1
2 principles.
• If bothPandQ are total, thenhP,Qiis total.
• If bothPandQ have finite tolerance, thenhP,Qihas finite tolerance.
Proof. Immediate.
Corollary 2.8. LetPandQbeΠ1
2statements, assume that both are total and that Phas finite tolerance, and letm≥1 be given.
(1) If hQ,Pmi ≤
sWPm thenQω≤sWPm.
(2) If hQ,Pmi ≤
WPm thenQω≤WPm.
Proof. Repeatedly applying Lemma 2.7, we see that Pm is total and has finite
tolerance. The result follows from the Squashing Theorem.
Corollary 2.9. Let Pbe a Π1
2 principle that is total and has finite tolerance, and
letm≥1 be given. (1) If Pm+1≤
sWPm, thenPω≤sWPm.
(2) If Pm+1≤
W Pm, then Pω≤WPm.
Proof. SincePm+1 ≤
sW Pm, we know that hP,Pmi ≤sW Pm, so the result follows
from the previous corollary.
For the remainder of this article, we employ the following short-hand to avoid excessive exponents and to givePωa more evocative name.
Statement 2.10. For any Π1
2 principle P, we denoteω applications ofP, or Pω,
bySeqP. We callSeqPthesequential versionof P.
So, for instance, Corollary 2.6says that that if Pis total and has finite tolerance, then P2 ≤sW P implies that SeqP ≤sW P. With this terminology, we have the
following simple result.
Proposition 2.11. Let PandQ beΠ1
2 principles.
(1) If P≤sWQ, thenSeqP≤sWSeqQ.
(2) If P≤WQ, thenSeqP≤WSeqQ.
Proof. For (1), fix Φ and Ψ witnessing the reductionP≤sWQ. Given an instance
3. Ramsey’s theorem for different numbers of colors
Throughout this section, letn≥1 be fixed. Our goal is to work up towards a proof of the following theorem.
Theorem 3.1. For allj, k≥2 with j < k, we haveRTnk sWRTnj.
As pointed out above, we have that RCA0 ⊢ RTnj → RTnk, but the obvious proof
uses multiple nested applications ofRTnj. Theorem3.1says that it is impossible to give a uniform proof of this implication using just one application ofRTnj.
The key ingredients of the proof are Proposition 2.1, the Squashing Theorem, and the fact that it is possible to code more intoSeqRTnk than intoRTnk alone. To illustrate the last of these, consider RT12. Notice that every computable instance ofRT12trivially has a computable solution because either there are infinitely many
0s or there are are infinitely many 1s (and each of these sets is computable), but there is one non-uniform bit of information used to determine which of these two statements is true. However, it is a straightforward matter to build a computable instance ofSeqRT12 such that every solution computes ∅′. The idea is to use each
column to code one bit of∅′ by exploiting this one non-uniform decision. In fact,
for higher exponents this result can be made sharper, as we now prove. (See also [24, Proposition 47] for a related result in the context of proof mining and program extraction.)
Lemma 3.2. There is a computable instance of SeqRTn2 every solution to which
computes∅(n).
Proof. We prove the result fornbeing odd; the case wherenis even is analogous. Fix a computable predicateϕsuch that
∅(n)={i∈ω: (∃x0)(∀x1)· · ·(∃xn
−1)ϕ(i, x0, x1, . . . , xn−1)}.
Define a computable sequence of coloringshfi:i∈ωiby
fi(y) =
(
1 if (∃x0< y0)(∀x1< y1)· · ·(∃xn−1< yn−1)ϕ(i, x0, x1, . . . , xn−1),
0 otherwise,
for ally=hy0, y1, . . . , yn−1i ∈[ω]n.
Let hHi : i∈ωi be any sequence of infinite homogeneous sets for the fi. We claim that ∅(n)(i) = fi([Hi]n) for all i, and hence that ∅(n) ≤
T hHi : i ∈ ωi. To
see this, suppose first that i ∈ ∅(n). Let hw2j : 2j < ni be Skolem functions for
membership in∅(n), so that
(∀x1)(∀x3)· · ·(∀xn−2)ϕ(i, w0(i), x1, w2(i, x1), x3, . . . , wn−1(i, x1, x3, . . . , xn−2)).
Now define an increasing sequencez0< z1<· · ·< zn−1 of elementsHi as follows.
Start by letting z0 be the least z ∈Hi that is greater than w0(i). Then, given j
with 1 ≤ j ≤ n−1, suppose we have defined zk for all k < j. If j is odd, let
zj be the least z ∈ Hi that is greater than zj−1. Ifj is even, let zj be the least z ∈Hi that is greater than zj−1, and also greater than wj(i, x1, x3, . . . , xj−1) for
all sequencesx1, x3, . . . , xj−1 withxk< zk for each oddk < j.
The sequence ofzj so constructed now clearly satisfies
(2) (∃x0< z0)(∀x1< z1)· · ·(∃xn−1< zn−1)ϕ(i, x0, x1, . . . , xn−1).
So by definition offi, we have thatfi(z0, . . . , zn−1) = 1. And since thezj all belong
Now suppose thati /∈ ∅(n). We can similarly construct a sequencez0<· · ·< zn
−1
of elements ofHi witnessing thatf([Hi]n) = 0. Lethw2j+1: 2j+ 1< nibe Skolem
functions for non-membership in∅(n), so that
(∀x0)(∀x2)· · ·(∀xn−1)¬ϕ(i, x0, w1(i, x0), x2, . . . , wn−2(i, x0, x2, . . . , xn−3), xn−1).
Letz0 be the least element ofHi, and suppose we are given ajwith 1≤j≤n−1 such thatzk has been defined for allk < j. Ifj is even, letzj be the least z∈Hi
that is greater thanzj−1. Ifj is odd, letzj be the leastz∈Hithat is greater than zj−1, and also greater thanwj(i, x0, x2, . . . , xj−1) for all sequencesx0, x2, . . . , xj−1
withxk< zk for each evenk < j.
This sequence ofzj satisfies the negation of (2) above, so fi(z0, . . . , zn−1) = 0
by definition. Since all thezj belong toHi, the claim follows.
After relativization and translation into the language of strong Weihrauch re-ductions, we obtain from the above that TJn ≤sW SeqRTn2. (See the discussion
following Corollary5.21for a definition of the iterated Turing jump,TJn.)
Lemma 3.3. For alln≥1 andk≥2, we havehRT2n,RTnkiWRTnk.
Proof. Suppose instead that hRTn2,RTnki ≤W RTnk. Since RTn2 and RTnk are both
total and have finite tolerance by2.3, we may use the Squashing Theorem 2.5 to conclude that SeqRTn2 ≤W RTnk. Fix Φ and Ψ witnessing the reduction, and let f =hfi:i∈ωibe any computable instance ofSeqRTn2. Apply Φ to this sequence to obtain an instancegofRTnk and notice thatgis computable. By Theorem 5.6 of Jockusch [21], we can find an infinite setHhomogeneous forgsuch thatH′≤
T ∅(n)
(since RT1k is computably true, Jockusch’s Theorem 5.6 holds also when n = 1). We then have thatS=hSi:i∈ωi= Ψ(f, H) is a solution tof =hfi:i∈ωiwith
S′≤
T ∅(n).
But as the sequence f = hfi : i ∈ ωi was chosen as an arbitrary computable instance of SeqRTn2, this would imply that every computable instance ofSeqRTn2
has a solution with jump computable in ∅(n). This contradicts Lemma 3.2, since
no such set can compute∅(n). Therefore, we must havehRTn 2,RT
n
kiWRTnk.
We shall prove Theorem 3.1 by means of the following weaker version of the theorem, which now follows easily.
Corollary 3.4. For alln≥1 andk≥2, we haveRTn2kWRTnk.
Proof. Suppose instead that RTn2k ≤W RTnk. We know from Proposition 2.1 that
hRTn2,RTnki ≤W RTn2k. Hence, using transitivity of≤W, we have hRTn2,RT n ki ≤W
RTnk, contrary to Lemma3.3.
In order to use this corollary to handle all cases of Theorem 3.1, we use the following result saying that we can fan out a strong Weihrauch reductionRTnk ≤sW RTnj to obtain a strong Weihrauch reduction with a larger spread between the
number of colors used.
Lemma 3.5. Let n, j, k, s≥1. IfRTnk ≤sWRTjn, thenRTnks ≤sWRTnjs.
Proof. Fix Φ and Ψ witnessing the fact thatRTnk ≤sWRTnj. In what follows, define e(b, a, i) for allb, a∈ωand alli <⌊logba⌋to be theith digit in the basebexpansion
Fix an arbitrary f: [ω]n → ks. We now convert f into s many colorings f0, . . . , fs−1: [ω]n→kby setting
fi(x) =e(k, f(x), i)
for all i < sand all x∈[ω]n. Then for any x, the expansion off(x) in basek is
preciselyf0(x)· · ·fs−1(x). Hence, any set that is simultaneously homogeneous for
each of thefi is also homogeneous forf.
Now apply the reduction Φ to eachfi to obtain coloringsgi: [ω]n→j for each i < s. We merge thesemmany colorings into one coloringg: [ω]n →jsdefined by
g=
s−1 X
i=0 jigi.
Notice that any infinite set H homogeneous forg is simultaneously homogeneous for each of thegi. Hence, Ψ(H) is simultaneously homogeneous for each of thefi. But then by the observation above, it follows that Ψ(H) is an infinite homogeneous set forf. Since the reduction fromf togwas uniformly computable, the lemma is
proved.
We can now prove our main result.
Proof of Theorem 3.1. Seeking a contradiction, fix j < k and assume RTnk ≤sW RTnj. Since kj >1, we may fixs∈ωwith (
k j)
s>4, so that 4js< ks. Letm∈ω be
least such thatjs≤2m. We then have 2m−1< js, so 2m+1<4js< ks, and hence js≤2m<2m+1< ks.
Since we are assuming RTnk ≤sW RTnj, we can use Lemma 3.5 to conclude that RTnks ≤sWRTnjs. We therefore have
RTn2m+1 ≤sWRTnks≤sWRTnjs ≤sWRTn2m
Since ≤sW is transitive, it follows that RTn2m+1 ≤sW RTn2m, contradicting
Corol-lary3.4.
It is worth pointing out that, in proving of Theorem3.1, the proof of Lemma
3.5was the only moment where it mattered that we were working with thestrong form of Weihrauch reducibility. Specifically, since Ψ there took solutions to gi to solutions tofifor eachi, in finding a simultaneous solutionH for all thegiwe found a simultaneous solution Ψ(H) for all the fi. This would no longer be the case if joining with original instances was permitted, since then we could not guarantee that Ψ(gi, Hi) would be the same set for alli. We do not know how to overcome this difficulty, and hence leave open the question of whether Lemma 3.5and Theorem
3.1also holds with≤sWreplaced by≤W.
4. Weak Weak K¨onig’s Lemma
As discussed in Section 2, it is straightforward to see that SeqWKL ≤sW WKL
(and the reverse direction is obvious). However, the situation of WWKLis more interesting. By performing the same interleaving process to show thatWKL2≤sW WKL, one checks that the resulting tree has positive measure if each of the two input trees do (in fact, the measure of the interleaved tree is the product of the measures of the original trees), and hence it follows thatWWKL2≤sWWWKL. By iterating
measure many paths, one can interleave them to obtain a tree S whose measure will be the product of the Ti (and hence also positive) such that from any path throughS, one can uniformly compute paths through the Ti. However, this idea does not carry over to the case of an infinite sequence of trees of positive measure, since then the interleaving process can produce a tree of measure 0. Indeed, this can happen even if the measures of the trees in the sequences are uniformly bounded away from 0.
Notice that we trivially have WWKL ≤sW WKL, so SeqWWKL ≤sW SeqWKL
by Proposition2.11. As explained in Section 2, we have SeqWKL≤sW WKL, and
henceSeqWWKL≤sWWKLby transitivity of≤sW. One can also show that this can
be formalized to giveRCA0 ⊢WKL→SeqWKL→ SeqWWKL. The next theorem shows that the converses are also true, and henceSeqWWKL, even in this weaker form, is in fact strictly stronger thanWWKL.
Theorem 4.1.
(1) WKL≤sWSeqWWKL.
(2) RCA0⊢SeqWWKL→WKL.
In fact, both of these statements hold even if we restrict SeqWWKL to infinite sequences of subtrees of2<ω of measure uniformly bounded away from0.
Proof. We prove (2) in the stronger form in order to handle the formalized version carefully, but our construction is completely uniform and hence can be turned into a proof of (1).
Let S be an arbitrary infinite subtree of 2<ω. We define a sequence of trees
hTσ : σ ∈2<ωiindexed by finite binary strings σ ∈2<ω (which of course can be
put in bijection withω). Intuitively, Tσ is constructed as follows. Put the empty string ∅, 0, and 1 inTσ. Keep building above both 0 and 1 putting in all possible extensions as long asσ0 andσ1 both look extendible inS. If we discover that one ofσ0 orσ1 is not extendible inS, then stop building above 0 or 1 inTσaccordingly, and forever build above the other side (even if the other also ends up not extendible inS). In this way,Tσ will always have measure either 12 or 1.
More formally, we define our sequence as follows. Given ρ ∈ 2<ω and k ∈ ω,
let ExtS(ρ, k) be the ∆0 predicate saying that either k ≤ |ρ|, or there exists an
element ofS extendingρ of lengthk. Givenσ∈ 2<ω, define Tσ to be ∅ together
with the set ofτ ∈2<ω\{∅}satisfying one of the following:
• τ(0) = 0 andExtS(σ0,|τ|).
• τ(0) = 1 andExtS(σ1,|τ|).
• τ(0) = 0 and (∃k <|τ|)[ExtS(σ0, k)∧ ¬ExtS(σ1, k)].
• τ(0) = 1 and (∃k <|τ|)[ExtS(σ1, k)∧ ¬ExtS(σ0, k)].
• (∃k <|τ|)[ExtS(σ0, k)∧ExtS(σ1, k)∧¬ExtS(σ0, k+1)∧¬ExtS(σ1, k+1)]. Note that the last condition handles the case when both sides die at the same level, and in this situation we (arbitrarily) build the full tree.
Since S is tree, if k < m and ExtS(ρ, m), then ExtS(ρ, k). By Σ0
1-induction
and the fact thatExtS(ρ,0) holds by definition, if¬ExtS(ρ, m) then there exists a uniquek∈ω with k < msuch thatExtS(ρ, k) and¬ExtS(ρ, k+ 1). Using these facts, it is straightforward to check that eachTσ is a tree, and that for eachm∈ω, either every element of 2m is inTσ or exactly half of the elements of 2mare inTσ.
Applying SeqWWKL to the sequence hTσ : σ ∈ 2<ωi, we obtain a sequence
C: ω → {0,1} recursively by letting C(n) = BC↾n(0), where C↾n is the finite
sequenceC(0)C(1)· · ·C(n−1). We claim thatCis a path throughS. To show this, we prove the stronger fact that for eachn∈ω, we have (∀m)ExtS(C↾n, m). The proof is by induction onn(using Π0
1-induction, which follows from Σ01-induction).
For n = 0, note that C↾ n = ∅, and we know that (∀m)ExtS(∅, m) because
S is an infinite tree by assumption. Suppose that we have a given n ∈ ω for which (∀m)ExtS(C↾n, m). In this case, at least one of (∀m)ExtS((C↾n)0, m) or (∀m)ExtS((C↾n)1, m) must hold. Now ifi∈ {0,1}is such that¬ExtS((C↾n)i, m), thenTC↾nhas no node extendingiof lengthm(by definition of theTσ), so it must
be the case thatBC↾n(0) = 1−i. Therefore, we must have (∀m)ExtS(C↾(n+1), m).
This completes the induction, and the proof.
Fact (1) above can also be derived from the result of Brattka and Gherardi [5, Theorem 8.2] thatWKL≤sWSeqLLPOand the observation of Brattka and Pauly [7,
Figure 1] that thatLLPO≤sWWWKL. (See [5, Section 1] for a definition ofLLPO.)
On the other hand, we have the following fact, which follows in this form from more general results of Brattka and Pauly [7, Proposition 22], and also essentially by the proof of Simpson and Yu [36] thatWWKL9WKLoverRCA0. We include a proof for completeness.
Proposition 4.2. WKLW WWKL.
Proof. By results of Jockusch and Soare [23, Theorem 5.3], there is a computable instance of WKLfor which only measure 0 many elements of 2ω compute a
solu-tion. However, every 1-random computes an infinite path through every infinite computable instance ofWWKL. (See, e.g., [1, Lemma 1.3].)
Thus while WWKLn ≤sW WWKLfor each n∈ω, we have thatSeqWWKLW WWKL. Notice that the Squashing Theorem does not apply toWWKLbecause it is not total (there is no clear way to view every real as coding an instance ofWWKL). We now turn to questions about uniformly passing back and forth between trees of positive measure. Consider any such tree T of 2<ω. A question that seems
natural is whether from a positive rational q < 1, it is possible to build a tree
S of measure at least q, each path through which computes a path through T. Intuitively, is it possible to blow up the measure of T without losing information about its paths? It is not difficult to see that the answer is affirmative, and in fact, that such anS can be obtained uniformly fromqand an index forT. Indeed, fix a universal Martin-L¨of test {Ui :i ∈ ω} and let S = 2<ω−Ui for the least i
withq≤1−2−i. Every path throughS is 1-random, and hence computes a path
throughT, but not uniformly. The following lemma and proposition show that if we allowS to be defined non-uniformly fromT andq, then we can arrange for the computations from paths to paths to be uniform.
Lemma 4.3. Given a treeT ⊆2<ω of positive measure p, and given ε >0, there
is a treeS, each path of which uniformly computes a path through T, such that the measure of the complement ofS is at most (1 +ε)(1−p)2.
Proof. We may assumep <1, since otherwise we can just takeS =T. Fix a positive
δ <1 such that 1−δp≤(1+ε)(1−p). Choose minimal, hence incompatible, strings
σ0, . . . , σn−1∈/ T such that X
i<n
and let
S=T ∪([
i<n σiT),
whereσiT ={σiτ :τ ∈T}. Then the measure of the complement ofS is
(1−p)−X
i<n
2−|σi|p≤(1−p)−δp(1−p) = (1−p)(1−δp)≤(1 +ε)(1−p)2.
Now let Φ be the functional that sendsA∈2ω toA(|σi|)A(|σi|+ 1)· · · ifσi A
for somei < n, and toA otherwise. Clearly, Φ(A) is a path throughT whenever
Ais a path throughS.
Proposition 4.4. Given a treeT ⊆2<ωof positive measurep, and given a positive
rational q < 1, there is a tree S ⊆2<ω of measure at least q, each path of which
uniformly computes a path throughT.
Proof. GivenT,p, andq, chooseε0, . . . , εn−1 so that
(3) (1 +εn−1)(1 +εn−2)2· · ·(1 +ε0)2(n−1)(1−p)2n<1−q.
Now iterate the lemma. Let S−1=T, and givenSi−1 obtainSi with complement
of measure at most (1 +εi)(1−µ(Si−1))2such that each path throughSicomputes
a path throughSi−1. By induction, the complement ofSn−1has measure bounded
by (3), and each path through it computes a path through T.
Thus, we can either uniformly blow up the measure of a given treeT, and have paths through the new tree non-uniformly compute paths through the old; or we can non-uniformly blow up the measure of T, and have paths through the new tree uniformly compute paths through the old. The following proposition, which is a direct corollary of Theorem 4.1, shows that we cannot achieve both types of uniformity simultaneously.
Proposition 4.5. There is no effective procedure that, given (an index for) a com-putable subtree T of 2<ω of positive measure, and a positive rational q, produces (an index for) a computable subtree S of 2<ω of measure at least q and an e∈ω
such that ΦA
e is a path through T for every pathAthrough S.
Proof. Suppose otherwise and fix any computable sequencehTi :i∈ωiof (indices for) subtrees of 2<ω of positive measure. We build a single tree S of positive
measure, every path through which computes a sequence of sets hAi:i∈ωisuch that eachAiis a path throughTi. In particular, every 1-random set computes such a sequence. Of course, this contradicts the proof of Theorem4.1, as it follows from what is shown there that there exists a sequence of trees for which only sets of PA degree can compute a sequence of paths, but not every 1-random computes a set of PA degree.
We obtain S by interleaving the members of a new sequence hSi : i ∈ ωi of subtrees of 2<ω, constructed inductively as follows. By adding a tree tohTi :i∈ωi
if necessary, we may assumeµ(T0)<1, and fix a positive rational numberr with
Clearly, the resulting sequenceshSi :i∈ωiand hei :i∈ωiare computable. It follows that S is computable, and by construction, µ(S) = Qi∈ωµ(Si) ≥ r >0. Now supposeB is any path throughS. By undoing the interleaving process along
B, we computably define a sequence hBi : i ∈ ωi such that each Bi is a path throughSi. SettingAi = ΦBi
ei for eachi, it follows thathAi :i∈ωiis the desired
sequence of paths through theTi.
The preceding results inspire the following restriction ofWWKL. Letq <1 be a positive rational.
Statement 4.6 (q-WWKL). Every subtree T of 2<ω such that
|{σ∈2n :σ∈T}|
2n ≥q
for allnhas an infinite path.
Note that Proposition 4.4 can be formalized to show that RCA0 ⊢ WWKL ↔
q-WWKL, for eachq. We conclude this section with the following contrasting result.
Proposition 4.7. For all positive rationals p < q <1,p-WWKLWq-WWKL.
Proof. Suppose not, and let Φ and Ψ witness a Weihrauch reduction fromp-WWKL
toq-WWKL. We build a computable treeTof measure at leastpsuch that Φ(T) has measure less thanq, and thus obtain the desired contradiction. Intuitively, we use the fact that Ψ must take paths through Φ(T) to paths throughTto successively cut down larger and larger portions of Φ(T) by cutting down larger and larger portions ofT. Although this results in the measures of both trees becoming smaller, we will only cut down each tree finitely many times, and we will be able to control for how much of the measure ofT is left.
We shall regard each partial computable function as defining an initial segment of a computable subtree of 2<ω, with each new convergence giving an entire new
level of the tree, and only strings of maximal length at the previous level being extended. Then Φ in the construction can be viewed as a monotone map between such initial segments. This will ensure that the construction ofT will be uniform, and so by the recursion theorem, we can fix an index for it ahead of time. This permits us the convenience of not needing to consider T in the oracle for Ψ, by replacing that functional, if necessary, byΨ(b X) = Ψ(T, X).
Construction. Fix a positive number a such that 2−a < q−p. At stage sof the
construction we shall defineTs=T∩2≤s, starting withT0={∅}. That is, Ts will
have heights. Letnsbe the height of Φ(Ts), and assume without loss of generality thatns≤sfor allsand that Φ(T0) ={∅}.
At stages+ 1, choose the leastamany numbers x0<· · ·< xa−1 that we have
not yet acted for, as defined below. Assume inductively that for eachα∈2a there
is a stringσ∈Ts of lengthswithσ(xj) =α(j) for allj withxj < s. We consider two cases.
Case 1. If any of the following apply:
• Φ(Ts) contains fewer than 2nsq many strings of lengthns;
• xa−1≥s;
then we obtainTs+1 fromTsby addingσ0 andσ1 for eachσ∈Tsof lengths. Case 2. Otherwise, choose α∈ 2a so that Ψ(τ)(xj) ↓=α(j) for all j < a for at
least 2−a many stringsτ ∈Φ(Ts) of lengthns. Then, we obtainTs+1 from Ts by
adding σ0 and σ1 for each σ∈Ts of length s withσ(xj)6=α(j) for some j < a. Say we haveacted forx0, . . . , xa−1.
Verification. Clearly,T is a computable subtree of 2<ω. Note that the measure of T is cut down only when the construction enters Case 2, at which point it is cut down by a factor of precisely 2−a. Likewise, whenever the construction enters Case
2, the measure of Φ(T) is cut down by at least a factor of 2−a. We claim there is a
stagessuch that Φ(Ts) contains fewer than 2nsq many strings of lengths, so that
the measure of Φ(T) is less thanq. Fix the least suchs. Then as 2−a < q−p, it
follows thatTscontains at least 2spmany strings of lengths. But the construction
can never enter Case 2 at any stage afters, so the measure of T is at leastp. It thus remains only to prove the claim. To this end, lettbe any stage such that Φ(Tt) contains at least 2ntqmany strings of lengthnt. Fix the leastx0<· · ·< xa
−1
not yet acted for prior to staget+ 1. For each pathB through Φ(T), we have that Ψ(B)(xj)↓for allj < a, so by compactness, there is ans >max{t, xa−1}such that
Ψ(τ)(xj)↓for allj < aand allτ∈Φ(Ts) of lengthns. Fix the least suchs. Then the construction never enters Case 2 strictly between stagestands, soTscontains at least 2nsq many strings of length ns, so Case 2 applies at stage s. Hence, we
have shown that the construction continues to enter Case 2 until the measure of Φ(T) has been sufficiently cut down, from which the claim follows.
5. The Thin Set Theorem
For all n ≥ 1 and k ∈ {2,3,4, . . . , ω}, say that a subset S of ω is thin for a coloringf: [ω]n→kif there exists ac < k such thatf(x)6=c for allx∈[S]n. In
this section, we shall concentrate on the following combinatorial principle, known as the Thin Set Theorem.
Statement 5.1 (TSnk). Letn≥1 and letk∈ {2,3,4, . . . , ω}. Everyf: [N]n →k
admits an infinite thin set.
The statement TSnω is the usual Thin Set Theorem as studied in [8].2 Note that
TSn2 is logically equivalent toRTn2, i.e., the thin sets for 2-colorings are precisely the homogeneous sets. Likewise, observe that whereas RTn1 is plainly true,TSn1 is not even defined above, as it would be plainly false.
Implications between versions of the Thin Set Theorem for different numbers of colors go opposite the way they do for Ramsey’s theorem. For the purpose of viewing TSnk as a multi-valued function, it is important that a solution to an
instance of TSnk includes which color is omitted by the thin set since there is no
uniformly computable way to recover that information from the thin set alone.
Proposition 5.2. Let n≥1.
(1) If j, k≥2with j < k, thenTSnk ≤sWTSnj.
2This should not be confused with the principleTSn
<∞, which, by analogy with Ramsey’s
theorem, should be defined as (∀k≥ 2) TSnk. By contrast,TSnω is the statement of the Thin
Set Theorem for coloringsf : [N]n →ω, i.e., colorings employing infinitely many colors. Using
Proposition5.2, is not difficult to see thatTSn<∞is equivalent toTS n
(2) If j, k≥2with j < k, thenRCA0⊢TSnj →TSnk. (3) If j≥2, thenTSnω≤sWTSnj.
(4) If j≥2, thenRCA0⊢TSnj →TSnω.
Proof. We prove (1) and (2) (the argument is uniform and can easily be formalized inRCA0). Let j, k≥2 withj < k. Fixf: [ω]n→k. Defineg: [ω]n→j by letting
g(x) =
(
f(x) iff(x)< j−1, j−1 otherwise
for all x ∈[ω]n. Now suppose S ⊆ω is an infinite thin set for g, say withc < j
such thatg(x)6=cfor allx∈[S]n. Ifc < j−1, thenf(x)6=c for allx∈S, while
if c =j−1 then f(x)< j−1 for all suchx, so in particular f(x)6= j−1 =c. Either way,cwitnesses thatSis an infinite thin set forf. The proof of (3) and (4) similarly proceeds by collapsing all colors greater thenj−1 to bej−1.
Thus, we have the following chain for anyn:
TSnω≤sW. . .≤sWTSn4 ≤sWTSn3 ≤sWTSn2 =RTn2 ≤sWRT3n≤sWRTn4 ≤sW. . .
By Theorem3.1, none of the reductions to the right of the equals sign reverse. We shall see in Theorem5.27that the same is true of the left side whenn= 1.
5.1. General reverse mathematics results. Before discussing uniform impli-cations and sequential forms, we prove several results about the principles TSnk. General questions about the strength ofTSnk were asked by J. Miller at theReverse Mathematics: Foundations and Applications Workshop in Chicago in November, 2009. Another recent investigation of these principles appears in Wang [32].
Proposition 5.3. For each m, n, k≥1, we haveRCA0⊢TSmn+1kn →TSm+1k .
Proof. The result is trivial forn= 1, so we may assumen≥2. Letf: [N]m+1→k
be a coloring. Defineg: [N]mn+1→kn by
g(x,y0, . . . ,yn−1) =hf(x,y0), . . . , f(x,yn−1)i
for allx∈ω andy0, . . . ,yn−1∈[ω]m withx <y0<· · ·<yn−1.
Suppose H is an infinite set that avoids the color ha0, . . . , an−1i < kn for the
coloringg. Choose the greatest i < n for which there are infinitely many x∈H
such that
(4) f(x,y0) =a0, . . . , f(x,yi) =ai
for some y0, . . . ,yi ∈[H]m with x < y0 <· · · <yi. By assumption on the color
avoided byH, it must be thati < n−1.
By choice of i, we can remove finitely many elements from H if necessary to ensure that if x < y0 <· · · < yi satisfy (4) above, then there is no y >yi such
thatf(x,y) =ai+1. LetH′ beH with these finitely many elements deleted.
Now using ∆01 comprehension, we can define a sequencehxj:j∈ωiof elements
ofH′ so that for eachj, (4) holds for some y0, . . . ,y
i∈[H′]mwith
(5) xj<y0<· · ·<yi< xj+1.
Let R ⊆ H′ be the range of this sequence, which exists because the sequence is
increasing. We claim thatRavoids the colorai+1forf. Indeed, supposef(x,y) =
ai+1 for some x ∈ R and y ∈ [R]m with x < y. Let y0, . . . ,y
witnesses for having chosenx to belong to our sequence. Then by (5), it follows thatyi <y, which contradicts the definition ofH′.
Settingk= 2, we obtain:
Corollary 5.4. For eachm, n≥1, we haveRCA0⊢TSmn+12n →RTm+12 .
The proof of Proposition 5.3 is not entirely uniform. Indeed, one assumes Σ0 2
-induction in order to prove that∀n(TSmn+1kn →TS
m+1
k ). Similarly, the proof does
not give a Weihrauch reduction ofTSm+1k toTSmn+1kn .
Since RT32 is equivalent to arithmetic comprehension, we also get that TS2n+12n
implies arithmetic comprehension for each n ∈ω. However, we can do better by carefully choosing the coloring.
Proposition 5.5. For each n≥1, we haveRCA0⊢TSn+22n →ACA.
Proof. We will show how to reduce finding the range of an injectionf :N→Nto an instanceg : [N]n+2→2n of TSn+2
2n . Namely, for each i < n, define the coloring
gi: [N]n+2→2 by
gi(x0, . . . , xn+1) =
(
1 (∃z)[xi < z < xi+1∧f(z)< x0],
0 (∀z)[xi < z < xi+1→f(z)≥x0],
and let g=hg0, . . . , gn−1i. Let H be an infinite set that avoids at least one color b = hb0, . . . , bn−1i < 2n. Let m < n be the largest index for which there are x0<· · ·< xn+1 inH withgi(x0, . . . , xn+1) =bi for alli < m, and assume without loss of generality that suchx0, . . . , xn+1 can be found in every tail ofH.
To determine whether some numberyis in the range off, choose some elements
x0 <· · · < xn of H with y < x0 and gi(x0, . . . , xn) = bi for i < m. Note that
f(z) ≥x0 for all z > xm, otherwise we could pick y0 =x0 <· · · < ym = xm < ym<· · ·< yn+1inH to realize at leastm+ 1 bits ofb, contradicting the choice of
m. Therefore,y is in the range off if and only ify∈ {f(0), . . . , f(xm)}.
By contrast, Wang [32, Theorem 3.1] has shown that for everyn, there is a k
such that TSnk does not implyACA overRCA0. Thus, the number of colors above
is important.
Proposition 5.6 (RCA0). For all n≥1, we haveRCA0⊢TSn+13n →RT1<∞.
Proof. By Proposition 5.3, it suffices to show that TS23 implies RT1<∞. Given
f:N→k, defineg: [N]2→3 by
g(x, y) =
0 iff(x) =f(y), 1 iff(x)> f(y), 2 iff(x)< f(y),
for allx < y. Suppose thatH is an infinite set that avoids one of the three colors. Note thatH cannot avoid the color 0, since otherwise the restriction off toH
would be an injection, which is impossible sinceH is infinite. So supposeH avoids the color 1, so that the restriction off toH is then non-decreasing. Any bounded non-decreasing function on an infinite set eventually stabilizes to a maximal value
m. Then f−1(m) is an infinite homogeneous set for f. The case whenH avoids
